GPT2: An improved model for tropospheric slantdelays in VLBI and GNSS analysis

Johannes Bohm1, Klemens Lagler2, Michael Schindelegger1, Hana Krasna1, Robert Weber1, and Gregor Moller1

1Vienna University of Technology, Vienna, Austriamail: johannes.boehm@tuwien.ac.at

2Eidgenossische Technische Hochschule, Zurich, Switzerland

Abstract—GPT2 is an improved empirical model forthe determination of tropospheric delays to be used inhigh-precision global analyses of space geodetic obser-vations at microwave frequencies, such as Global Navi-gation Satellite Systems (GNSS) and Very Long BaselineInterferometry (VLBI). It provides pressure, temperature,temperature lapse rate, water vapor pressure values, aswell as hydrostatic and wet mapping functions coefficients.The underlying horizontal resolution is 5 degrees, andthe parameters contain annual and semiannual variations.We show results of geodetic VLBI which demonstratethe improvement with GPT2 compared to earlier empiricalmodels for the tropospheric delays. Future extensions ofGPT2 will contain an improved parameterization for thecalculation of zenith wet delays.

BIOGRAPHIES

Johannes Bohm is Professor for Space GeodeticTechniques at the Vienna University of Technology. In hisPhD studies and thereafter, he dealt with troposphericpath delays and developed tropospheric models for theanalysis of observations from space geodetic techniques.His main interests now are atmospheric effects in spacegeodesy in general and the determination of global ter-restrial and celestial reference frames from VLBI obser-vations.

Klemens Lagler is Research and Teaching Assistantat the ETH Zurich. As the main part of his master thesisat the Vienna University of Technology he developed thenew tropospheric model GPT2 and evaluated it with otherexisting models. Now he is doing his PhD at the Instituteof Geodesy and Photogrammetry at the ETH Zurich.

Michael Schindelegger is PhD student at the Depart-ment of Geodesy and Geoinformation with a researchfocus on modeling atmospheric and non-tidal oceaniceffects on Earth rotation on different time scales. He hasgood experience in processing meteorological data fromnumerical weather models as needed for various geodeticpurposes.

Hana Krasna is a PostDoc scientist, working in theresearch group Advanced Geodesy at the Vienna Univer-sity of Technology. Her PhD thesis focused on the deter-mination of geodynamic parameters from VLBI, such ascomplex frequency-dependent Love and Shida numbers

of solid Earth tides and parameters of Free Core Nutation.Hana Krasna is member of the VLBI group and one ofthe main developers of the Vienna VLBI Software VieVS.

Robert Weber is associate professor at the Depart-ment of Geodesy and Geoinformation, Vienna Universityof Technology, Austria. His main fields of research areGlobal Navigation Satellite Systems, geodetic referencesystems, active GNSS reference station networks andapplications of GNSS for geodynamics and meteorology.Robert Weber served within several activity fields of theIGS, e.g. as the former chair of the GNSS Working Group.He teaches courses in satellite navigation and gravity fieldmodeling at TU Vienna and is involved in several projectssetting up regional GNSS reference networks, Network-RTK, Real-Time-PPP as well as GNSS Atmosphere Mon-itoring.

Gregor Moller is a second year PhD student at ViennaUniversity of Technology. In his Master thesis he anal-ysed a real-time GNSS network concerning quality andstability. His current research focuses on GNSS signalanalysis and tropospheric delay modelling - especially interms of meteorology and navigation application. Since2011, Gregor Moller is a member of the IGS TroposphereWorking Group.

I. INTRODUCTION

Tropospheric delay modeling is the major error sourcein the analysis of observations from space geodetictechniques, such as Global Navigation Satellite Systems(GNSS) or geodetic Very Long Baseline Interferometry(VLBI). For global applications aiming at highest preci-sion, such as geodynamical studies (cf. Tesmer et al.,2011 [12]) or reference frame solutions (cf. Tesmer et al.,2007 [11]), tropospheric delays ∆L are usually modeledas sum of hydrostatic and wet delays (see Equation1). Herein, the zenith hydrostatic delays ∆Lz

h are veryaccurately determined from pressure values at the siteapplying the model by Saastamoinen (1972 [10]) asrefined by Davis et al. (1985 [5]). Those values are thenmapped down to the elevation of the observation withthe hydrostatic mapping function mfh(e). On the otherhand, the zenith wet delays ∆Lz

w are estimated, e.g. aspiecewise linear offsets every 30 minutes, with the wetmapping functions mfw(e) as partial derivatives.

Figure 1: Amplitudes of annual pressure variations in hPa as providedwith GPT2

∆L = ∆Lzh ·mfh(e) + ∆Lz

w ·mfw(e) (1)

The Conventions of the International Earth Rotationand Reference Systems Service 2010 (IERS Conventions2010; Petit and Luzum, 2010 [8]) and their electronicupdates recommend using pressure values recorded atthe sites and the Vienna Mapping Functions 1 (VMF1;Bohm et al., 2006a [1]). If those values are not available,the analyst is advised to apply the empirical model GPT2(Lagler et al., 2013 [7]) or the earlier Global Pressure andTemperature model (GPT; Bohm et al., 2007 [3]) and theGlobal Mapping Functions (GMF; Bohm et al., 2006b [2]).

In Section II we review some properties of GPT2. Wediscuss its application in VLBI analysis in Section III andfinally provide an outlook in Section IV, describing thepotential application of GPT2 for navigation purposes.

II. PROPERTIES OF GPT2

The development and validation of GPT2 as well asthe comparison with GPT/GMF have been described indetail by Lagler et al. (2013 [7]). The output parametersof GPT2 are pressure, temperature, temperature lapserate, water vapor pressure, as well as hydrostatic and wetmapping function coefficients. These mapping functioncoefficients have to be used with VMF1 subroutines.GPT2 is an empirical (blind) model based on grids with ahorizontal resolution of 5 degrees, and every parameteris represented by annual and semiannual amplitudes andphases. In earlier models like GPT, GMF, or the ESA blindmodel (Kruger et al., 2004), there is no semiannual termand with GPT and GMF the phase of the annual term isfixed to January 28. For example, Figures 1 and 2 showthe amplitudes of the annual pressure variations and themonth of the annual pressure maximum. It is interestingto note that the largest annual pressure variations occurin Asia. However, the corresponding maxima and minimaare not always clearly allocated to the end of Januaryor July. Similar plots are provided for the temperature(Figures 3 and 4) and the specific humidity (Figures 5and 6) where the distribution of amplitudes and phasesis less varied but follows a clear annual pattern.

Figure 2: Month of maximum of the annual pressure variation in GPT2

Figure 3: Amplitudes of annual temperature variations in oC as providedwith GPT2

Figure 4: Month of maximum of the annual temperature variation inGPT2

In Figures 7, 8, and 9, we show zenith hydrostaticdelays, hydrostatic mapping functions, and wet mappingfunctions at station Tsukuba in Japan from 2010.0 to2012.0. In particular we show the values from GPT/GMF,GPT2, and from the Vienna Mapping Functions 1. To-gether with the mapping function coefficients of theVMF1, also zenith hydrostatic and wet delays are pro-vided which are determined from ray-tracing throughpressure level data of the European Centre for Medium-

Figure 5: Amplitudes of annual specific humidity variations in g/kg asprovided with GPT2

Figure 6: Month of maximum of the annual specific humidity variationin GPT2

2010 2010.5 2011 2011.5 2012

2.25

2.3

2.35

2.4

Year

Zen

ith h

ydro

stat

ic d

elay

s in

m

ECMWFGPTGPT2

Figure 7: Zenith hydrostatic delays at Tsukuba, Japan

Range Weather Forecasts (ECMWF). We find a goodagreement of all three models. We can also note thatthe additional semiannual term and the flexible phase ofthe annual variation support an even better agreement ofGPT2 with VMF1. Of course – as GPT/GMF and GPT2are blind models – they cannot account for the daily andweekly variations as available with the VMF1.

2010 2010.5 2011 2011.5 201210.08

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10.12

10.14

10.16

10.18

Year

Hyd

rost

atic

map

ping

func

tion

at 5

deg

rees

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Figure 8: Hydrostatic mapping functions at Tsukuba, Japan at 5degrees elevation

2010 2010.5 2011 2011.5 201210.5

10.6

10.7

10.8

10.9

11

Year

Wet

map

ping

func

tion

at 5

deg

rees

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Figure 9: Wet mapping functions at Tsukuba, Japan at 5 degreeselevation

III. APPLICATION IN VLBI ANALYSIS

We ran three global VLBI solutions with all observa-tions from 1984.0 to 2012.5 using the Vienna VLBI Soft-ware (VieVS; Bohm et al., 2012 [4]). We applied VMF1with pressure values recorded at the sites, GPT/GMF,and GPT2 for the three solutions. We followed the IERSConventions 2010 (Petit and Luzum, 2010 [8]), apart fromthe fact that we also applied non-tidal atmospheric load-ing corrections as provided by the NASA Goddard Group(Petrov and Boy, 2004 [9]). This procedure is importantfor studies of tropospheric delay models, because other-wise there would be a destructive effect between zenithhydrostatic delays and atmospheric loading (Tregoningand Herring, 2006 [13]). We determined the annual andsemiannual station height differences of the solutions withGPT/GMF and GPT2 with respect to the solution withVMF1 and local pressure values. We found an averageimprovement of 40% for the annual and the semiannualheight differences (see Figure 10) with GPT2 comparedto GPT/GMF.

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Figure 10: Difference in annual (upper plots) and semiannual (lowerplot) height amplitudes at stations included in more than 50 sessionswith GPT2 and GPT/GMF compared to VMF1 and pressure valuesrecorded at the sites (”NGS”)

IV. OUTLOOK

We plan to equip GPT2 with an improved capabilityfor calculating zenith wet delays. In its present version,GPT2 only provides pressure, temperature, temperaturelapse rate, and water vapor pressure values, so that wecould use the model by Saastamoinen (1972 [10]) (seeEquation 2) to determine zenith wet delays. In Equation2 the zenith wet delay ∆Lz

w is in meters, the water vaporpressure e in hPa, and the temperature T in Kelvin. Figure11 shows the zenith wet delays at station Tsukuba asdetermined from GPT2 and as derived from ray-tracingthrough pressure level data from the ECMWF.

∆Lzw = 0.002277 · ((1255/T + 0.05) · e) (2)

2010 2010.5 2011 2011.5 20120

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0.2

0.3

0.4

0.5

Year

Zen

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et d

elay

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ECMWFGPT2

Figure 11: Zenith wet delay at Tsukuba, Japan

In future, water vapor lapse rate and mean tempera-ture should also be added as output parameters, becausethen more sophisticated models could be applied for thezenith wet delays (see Kruger et al., 2004 [6]). For the

sake of improved zenith wet delays, we probably have touse a higher grid resolution than 5 degrees.

ACKNOWLEDGMENT

The authors would like to thank the Austrian ScienceFund (FWF) for funding projects GGOS Atmosphere(P20902-N10) and Integrated VLBI (P23143-N21).

REFERENCES

[1] Bohm, J., B. Werl, and H. Schuh (2006a), Troposphere map-ping functions for GPS and Very Long Baseline Interferome-try from European Centre for Medium-Range Weather Fore-casts operational analysis data, J. Geophys. Res., 111, B02406,doi:10.1029/2005JB003629.

[2] Bohm, J., A. Niell, P. Tregoning, and H. Schuh (2006b), GlobalMapping Function (GMF): A new empirical mapping function basedon numerical weather data, Geophys. Res. Lett., 33, L07304,doi:10.1029/2005GL025546.

[3] Bohm, J., R. Heinkelmann, and H. Schuh (2007), Short Note: Aglobal model of pressure and temperature for geodetic applications,J. Geod., 81(10), 679–683, doi:10.1007/s00190-007-0135-3.

[4] Bohm, J., S. Bohm, T. Nilsson, A. Pany, L. Plank, H. Spicakova,K. Teke, and H. Schuh (2012), The new Vienna VLBI software, In:Proceedings of the 2009 IAG Symposium, International Associationof Geodesy Symposia, Vol. 136, edited by S. Kenyon, M. C. Pacino,and U. Marti, pp. 1007–1012, ISBN 978-3-642-20337-4.

[5] Davis, J. L., T. A. Herring, I. I. Shapiro, A. E. E. Rogers, andG. Elgered (1985), Geodesy by radio interferometry: Effects ofatmospheric modeling errors on estimates of baseline length,Radio Sci., 20(6), 93–1607, doi:10.1029/RS020i006p01593.

[6] Kruger, E., T. Schuler, G. W. Hein, A. Martellucci, G. Blarzino(2004), Galileo Tropospheric Correction Approaches DevelopedWithin GSTB-V1, In: Proceedings of the GNSS 2004 - EuropeanNavigation Conference, 17-19 May 2004, Rotterdam, The Nether-lands.

[7] Lagler, K., M. Schindelegger, J. Bohm, H. Krasna, T. Nilsson(2013), GPT2: Empirical slant delay model for radio space geodetictechniques, Geophys. Res. Lett., doi:10.1002/grl.50288.

[8] Petit, G., and B. Luzum (2010), IERS conventions 2010, IERSTechnical Note, 36, 179 pp.

[9] Petrov, L., and J. P. Boy (2004), Study of the atmospheric pressureloading signal in very long baseline interferometry observations, J.Geophys. Res., 109, B03405, 1–14, doi:10.1029/2003JB002500.

[10] Saastamoinen, J. (1972), Atmospheric correction for the tropo-sphere and stratosphere in radio ranging of satellites, In: TheUse of Artificial Satellites for Geodesy, Vol. 15, edited by S. W.Henriksen et al., pages 247–251, AGU, Washington, D.C.

[11] Tesmer, V., J. Bohm, R. Heinkelmann, and H. Schuh (2007), Effectof different tropospheric mapping functions on the TRF, CRF andposition time-series estimated from VLBI, J. Geod., 81(6-8), pp.409-421, doi:10.1007/s00190-006-0126-9.

[12] Tesmer, V., P. Steigenberger, T. van Dam, and T. Mayer-Gurr(2011), Vertical deformations from homogeneously processedGRACE and global GPS long-term series, J. Geod., 85, 291–310.

[13] Tregoning, P., and T. A. Herring (2006), Impact of a priorizenith hydrostatic delay errors on GPS estimates of stationheights and zenith total delays, Geophys. Res. Lett., 33, L23303,doi:10.1029/2006GL027706.